Let \(\tilde{\Pi }^{(N,n)}\in {P}_{n}\)  be paintbox constructed from \(\tilde{\rho }^{(N)}\) . Proposition 2.9 in [1]} readily gives,
for any \(\tilde{\pi }\in {P}_{n}\) ,
\(\mathbb {P}\left(\tilde{\Pi }^{(N,n)} = \tilde{\pi }\right) = \mathbb {P}\left(\tilde{\Pi }^{(\infty ,n)} = \tilde{\pi }\right) + o\left(1\right),\) 
